Kronecker's First Limit Formula, Revisited 1

KRONECKER’S FIRST LIMIT FORMULA, REVISITED

W. DUKE, O.¨ IMAMOGLU,¯ AND A.T´ OTH´

To Don Zagier, with admiration

Abstract. We give some new applications of Kronecker’s first limit formula to real quadratic fields. In particular, we give a surprising geometrical relationship between the CM points associated to two imaginary quadratic fields with discriminants d and d′ and certain winding number functions coming from the closed geodesics associated to the real quadratic field of discriminant d′d.

1. Introduction The Kronecker limit formulas have been studied intensively and have inspired several generations of mathematicians since Kronecker’s paper [30] of 1863. Weil devoted a book [52] to their historical development. Siegel’s lucid treatment [44] makes their significance in number theory, especially in the study of L-functions, apparent. They directly influenced works by Weber [50], Lerch [34], Landau [32], Hecke [21], Herglotz [25], Chowla and Selberg [7], Ramachandra [37], Siegel [45], Stark [47], Ray and Singer [38], Zagier [54] and Shintani [42], to list chronologically a dozen prominent ones published at least forty years ago. The last forty years has mostly seen a (substantial) development of various generalizations of the classical limit formulas and the application of these generalizations in number theory, geometry and physics. In this paper we confine our attention almost exclusively to results closely connected to Kronecker’s classical first limit formula (KLF for short). After stating KLF, we will illustrate its application in some very special (and pretty) cases. Then we will sketch its proof and recall its relation to certain L functions for quadratic fields. This build-up is to motivate and place into context some new results we will present in the real quadratic case. In a sense this paper is a companion piece to [15], where other new results on geometric invariants associated to real quadratic fields can be found.

Acknowledgement: Duke and T´othare grateful to FIM of ETH Z¨urich for its generous continued support of our joint research.

2. The first limit formula Kronecker’s ﬁrst limit formula is a two dimensional version of a familiar one for the Riemann ∑∞ zeta function ζ(s) = n−s at s = 1: n=1 ( ) − 1 (1) lim ζ(s) − = γ0, s→1+ s 1

Date: March 30, 2018. Duke’s research on this paper was supported by NSF grant DMS 1701638, the Simons Foundation and the Mathematisches Forschungsinstitut Oberwolfach. A.´ Tóth is supported by NKFIH (National Research, Development and Innovation Office) grant ERC HU 15 118946. 1 2 W. DUKE, O.¨ IMAMOGLU,¯ AND A.T´ OTH´ where γ0 = 0.577216 ... is Euler’s constant. Let Q(x, y) = ax2 + bxy + cy2 be a real binary quadratic form with discriminant D = b2 −4ac. In case Q is positive definite, that is a > 0 and D < 0, Kronecker obtained a limit formula analogous to (1) for the zeta function ∑′ −s (2) ζQ(s) = Q(m, n) . m,n∈Z Here the prime indicates that (0, 0) is omitted from the sum. Since Q is positive definite, the sum defining ζQ(s) converges absolutely for Re s > 1, yet it blows up at s = 1, as does the Riemann zeta function. Kronecker’s limit formula has the same shape as (1), except that on the right hand side we have a term that depends on Q in a non-trivial way. This dependence is necessarily modular in that ζQ(s) = ζγQ(s) for γ ∈ Γ = PSL(2, Z) with γQ arising from Q by a unimodular change of variables. When the form Q is represented by √ −b + D z = ∈ H, Q 2a H the upper half-plane, it can be checked that zγ−1Q = γzQ where γ ∈ Γ acts as usual on z ∈ H as a linear fractional map. The modular function that appears is (3) H(z) = − log(y|η(z)|4) (H for height) where, for q = e(z) = exp(2πiz), ∏ η(z) = q1/24 (1 − qm) m≥1 is the Dedekind eta function. This H is a counterpart to the usual modular j-function j(z) = q−1 + 744 + 196844q + ··· . Although it is non-holomorphic, H is bi-harmonic with respect to the hyperbolic Laplacian.

Figure 1. The function H(z)

Theorem 1 (KLF). For a > 0 and D < 0 we have ( ) s ( ) | | 2 − 2π − (4) lim D ζQ(s) − = 2π 2γ0 log 4 + H(zQ) , s→1+ s 1 where γ0 is Euler’s constant. KRONECKER’S FIRST LIMIT FORMULA, REVISITED 3 Some examples Before describing a proof, here are some attractive classical applications of KLF that do not require a lot of preparation to appreciate. All of these examples involve an interplay between ζQ(s) and Dirichlet L-functions. Consider for a prime q ≡ 3 (mod 4) the positive deﬁnite integral quadratic form 2 − q+1 2 Q(x, y) = x xy + 4 y whose discriminant is D = −q. For a positive integer n plot the ellipse in R2 determined by the equation Q(x, y) = n. It is a lovely fact that for some q we have a simple exact formula

Figure 2. 2 − q+1 2 Ellipses x xy + 4 y = n for q = 7, 23, 31 and n = 2 2

-1

-2 -2 -1 0 1 2

2 as a function of n for the number rq(n) of lattice points in Z that lie on the ellipse: ∑ ( ) d (5) rq(n) = rq(1) q , d|n ( ) · where q is the Legendre symbol. Since the nineteenth century it has been known that formula (5) holds for all n when q ∈ {3, 7, 11, 19, 43, 67, 163}. Unfortunately, (5) does not work for all q. It obviously does for n = 1. For n = 2 the right hand side of formula (5) gives the value 4 when q ≡ 7 (mod 8) and Figure 2 illustrates that this works when q = 7 since there are 4 lattice points on the outermost ellipse, which is given by x2 − xy + 2y2 = 2. However, it fails for any larger q ≡ 7 (mod 8) as the resulting narrowing ellipses (illustrated for q = 23 and 31) cannot possibly contain any lattice points. It is instructive to try to extend this method to show that for any ﬁxed q ≡ 3 (mod 8) other than q = 3, 11, 19, 43, 67, 163, formula (5) must fail for some integer n > 0. For ﬁxed q ∈ {3, 7, 11, 19, 43, 67, 163} we can translate (5) into the form

(6) ζQ(s) = rq(1)ζ(s)L−q(s), where for any fundamental discriminant D the Dirichlet L-function is defined by ∑ −s LD(s) = χD(n)n , n≥1 4 W. DUKE, O.¨ IMAMOGLU,¯ AND A.T´ OTH´ ( ) · D with χD( ) = · the Kronecker symbol. Here we have applied quadratic reciprocity. Of course rq(1) = 2 unless q = 3, when r3(1) = 6. To avoid writing rq(1) we now leave out the case q = 3 and also, for the rest of this section, write L(s) for LD(s), as long as the value of D is understood. √π For these forms Q, KLF together with (1) and (6) implies both the evaluation L(1) = q and the deeper fact that √ √ q ′ − 1 1+ −q (7) π L (1) = γ0 2 log 4q + H( 2 ). One way to go further with (7) is to utilize the Euler product ∏ ( ) − p −s −s L(s) = (1 q p ) . p prime √ q ′ | Write π L (1) = D log L(s) s=1 in (7) and observe that it implies ∑ ( ) √ √ p log p − 1+ −q − π (8) q p = log q H( 2 ) + O(1) = 6 q + log q + O(1). p prime √ − π Roughly speaking, since the negative term 6 q on the RHS of (8) dominates, this as- ymptotic formula indicates that for (6) to hold for large q, there must be many quadratic non-residues modulo q among the first few primes. For instance, the first 12 primes are non- residues modulo 163. This is one way to become convinced that formula (5) cannot remain true for large q, since the resulting imbalance in the distribution of quadratic residues should contradict the generalized Riemann hypothesis. A different kind of application of KLF to (7) makes use of the identity − √ ∑q 1 ( ) ( ) q ′ − n n (9) π L (1) = γ0 + log 2π q log Γ q , n=1 which was derived in 1883 by Berger [2] and independently by Lerch [34] in 1897 using Kummer’s formula ( ) ∑ (10) log √1 Γ(x) = ( 1 − x)(γ + log 2π) + 1 log(2 sin πx) + 1 n−1 log n sin(2πnx). 2π 2 0 2 π n≥1 Taken together, (9) and (7) give for q = 7, 11, 19, 43, 67, 163 some remarkable evaluations: − ( √ ) ∏q 1 ( ) − 4 ( n ) 1+ q 1 n q (11) η 2 = 2πq Γ q , n=1 also due to Lerch [34]. Chowla and Selberg [7, 8] later independently derived (11) and applied it to evaluate elliptic integrals having singular moduli, a connection already glimpsed by Landau in 1902 [32, p.313]. An equivalent formulation can be given in terms of certain hypergeometric series: ∑ (a) (b) xn F (a, b, c; x) = 1 + n n , (c) n! n≥1 n where (a)n = a(a + 1) ··· (a + n − 1). Their result can be put into an elegant form using the Schläflimodular function η( τ+1 ) (12) f(τ) = e(− 1 ) 2 . 48 η(τ) KRONECKER’S FIRST LIMIT FORMULA, REVISITED 5 √ For q = 7, 11, 19, 43, 67, 163 and f = f( −q) it can be shown that (11) yields ∏q−1 ( ) − − 1 1 ( n ) 1 1 6 24 2 2 n 2 q (13) F ( 4 , 4 , 1; 2 f ) = (2qπ) f Γ q . n=1 √ √Weber (see Tabelle VI. of [51]) computed explicitly all these values of f. Thus f( −7) = 2 and the others are algebraic integers of degree 3 over Q that may be expressed in the (shockingly) simple form √ √ √ 1 1 3f( −q) = αq + (βq + 3 3q) 3 + (βq − 3 3q) 3 , where αq = 2 except that α19 = 0 and β11 = 17, β19 = 27, β43 = 35, β67 = 53, β163 = 135. In fact, diophantine properties of f are crucial in “resurrections” of Heegner’s proof ([24]) that 163 is the largest q for which (5) holds. See Stark’s ICM paper [46] for such a proof and also [48] for his recent description of some of the drama surrounding this result. See also Birch’s papers [3], [4] and that of Deuring [12]. Weber’s table of values of f also gives that for q = 5, 13, 37 √ √ 1 − 8 (14) 2 f( q) = t + u q, where (t, u) = (3, 1), (11, 3), (146, 24), respectively. It turns out that these (t, u) solve the Pell equation t2 − qu2 = 4 with t minimal among all solutions with t, u > 0. These give samples of Kronecker’s “solution” of the Pell equation using modular functions. The connection with KLF comes from the fact that for 2 2 ′ 2 − q+1 2 Q(x, y) = x + qy and Q (x, y) = 2x 2xy + 2 y when q = 5, 13, 37 we have that

2L−4(s)Lq(s) = ζQ(s) − ζQ′ (s). Now apply KLF to the right hand side and the famous result of Dirichlet √ √ 1 qLq(1) = log 2 (t + u q) to the left hand side to get (14).

3. Proof of KLF There are a number of proofs, starting with Kronecker’s own. See [52] for a discussion. Here we will give a brief treatment of one of the most transparent ones. For z ∈ H and Re s > 1 deﬁne E(z, s) by ∑′ 1 s | |−2s (15) ζ(2s)E(z, s) = 2 y m + nz . m,n∈Z We may work with ζ(2s)E(z, s) to prove KLF since s s+1 (16) |D| 2 ζQ(s) = 2 ζ(2s)E(zQ, s). The key input is the Fourier expansion of ζ(2s)E(z, s), which was found by Deuring [11] and Chowla-Selberg [8]. We simply state the result; an excellent source for its proof is [33]. It is convenient to state the result for the completion (17) E∗(z, s) = π−sΓ(s)ζ(2s)E(z, s). 6 W. DUKE, O.¨ IMAMOGLU,¯ AND A.T´ OTH´

− s As usual set Λ(s) = π 2 Γ( s )ζ(s). Then we have the Fourier expansion 2 ∑ ∗ s 1−s 1/2 1 −s (18) E (z, s) = Λ(2s)y + Λ(2 − 2s)y + 4y n 2 σ2s−1(n)K − 1 (2πny) cos(2πnx), s 2 n≥1 ∑ s where σs(n) = d|n d and Ks(y) is the usual Bessel function. By making use of (16) and (17), the proof of KLF can be reduced to the following statement: 1 (19) 2E∗(z, s) = + γ − log(4πy) − 4 log |η(z)| + O(s − 1). s − 1 0 To obtain (19), ﬁrst note that √ π −y K 1 (y) = e . 2 2y By (18) it follows that 1 ( ∑ ) 2E∗(z, s) = + γ − log(4πy) + 1 πy + 4Re n−1σ (n)qn + O(s − 1). s − 1 0 3 1 n≥1 Now apply ( ∑ ) ( ∑ ) ( ) −1 n −1 mn − | | πy Re n σ1(n)q = Re n q = log η(z) + 12 n≥1 m,n≥1 to ﬁnish the proof of (19), hence of KLF.

Remark: A quite diﬀerent kind of proof was given by Shintani [43] using the Barnes double gamma function.

Expansions around s = 0 It turns out to often be advantageous to have expansions of KLF–type around s = 0. An added bonus of the Fourier expansion (18) is that it renders as obvious the analytic ∗ continuation and functional equation of E (z, s). Using that Λ(1−s) = Λ(s), Ks(y) = K−s(y) and s −s n σ−2s(n) = n σ2s(n) it follows from (18) that s(s − 1)E∗(z, s) is entire and that (20) E∗(1 − s, z) = E∗(z, s).

This makes it easy to compute the Laurent expansion of ζQ(s) around s = 0, which no longer contains the mysterious Euler constant that occurs in the formulas around s = 1. By the functional equation (20) and (19) we have (21) 2E∗(z, s) = −s−1 + γ − log(4π) + H(z) + O(s) hence ( 0 ) 2 − 1 2 (22) ζQ(s) = −1 − log(8π |D| 2 ) − H(zQ) s + O(s ). There is a connection of E(z, s) to the spectral theory of the Laplacian on Γ\H hinted at by the occurrence of the Bessel function in the Fourier expansion (18), one that is somewhat miraculous with hindsight. In fact we can write ∑ (23) E(z, s) = (Im γz)s,

γ∈Γ∞\Γ KRONECKER’S FIRST LIMIT FORMULA, REVISITED 7

1 1 where Γ∞ is the usual parabolic subgroup generated by ( 0 1 ). Let 2 2 2 (24) ∆ = y (∂x + ∂y ) be the hyperbolic Laplacian. Now ∆Imz = s(s−1)Imz and ∆ commutes with linear fractional action. It follows that E(z, s) is an eigenfunction of −∆ : (25) −∆E(z, s) = s(1 − s)E(z, s). 1 For Re s = 2 it happens that E(z, s) gives the continuous part of the spectral decomposition of ∆ on Γ\H. Consider the following expansion of E(z, s) around s = 0: 2 (26) E(z, s) = H0(z) − H1(z)s + H2(z)s − · · · .

From (21) and (17) we see that H0(z) = 1 and H1 = H. Furthermore, upon using (25) in (26), we also have that ∆H = 1 and

∆Hn = Hn−2 + Hn−1, for n ≥ 2. In particular, H is bi-harmonic with respect to ∆. For related work see [31].

L-functions with genus characters Perhaps the most important applications of Kronecker’s limit formulas are to L-functions associated to quadratic fields and their abelian extensions. The first limit formula applies to abelian extensions that are unramified. In fact, we will only consider L-functions associated to unramified quadratic extensions, namely those with genus characters, for which there is already an extremely rich theory. √ Suppose that D ≠ 1 is a fundamental discriminant and K = Q( D). Let σ : K → K K Q ∈ K σ + generate the Galois group of / and for α let N(α) = αα . Let ClD be the group of (narrow) fractional ideal classes in K. Thus two ideals a and b are in the same class if there is α ∈ K with N(α) > 0 so that a = (α)b. Let + h(D) = #ClD be the class number and w = wD be the number of roots of unity in K so that w = 2 unless D = −3, −4 when w = 6, 4, respectively. If D > 1 let ϵD be the smallest unit of norm 1 that is > 1 in the ring of integers OK of K. Associated to an ideal class A is the partial zeta function ∑ −s ζK(s, A) = ζ(s, A) = N(a) , a where a runs over all non-zero integral ideals in A. Note that ζ(s, A) = ζ(s, A−1). Dirichlet applied his geometric method to evaluate { 2π s if D < 0 (27) lim (s − 1)|D| 2 ζ(s, A) = w → + s 1 log ϵD if D > 1. . Given a character χ of Cl+ we have the L-function D ∑ ∑ (28) L(s, χ) = χ(a)N(a)−s = χ(A)ζ(s, A).

a⊂OK ∈ + A ClD A genus is an element of the group of genera, which is (isomorphic to) the quotient group K + + 2 (29) Gen( ) = ClD/(ClD) . 8 W. DUKE, O.¨ IMAMOGLU,¯ AND A.T´ OTH´

K ∼ Z Z ω(D)−1 + It is known that Gen( ) = ( /2 ) so if GD is a genus in ClD then 1−ω(D) (30) #GD = 2 h(D), where ω(D) is the number of distinct prime factors of D. The characters of Gen(K) descend + from the real characters of ClD and are in one to one correspondence with co-prime factor- izations D = d′d where d and d′ are fundamental discriminants (including 1). Kronecker discovered a factorization of L(s, χ) for such a χ that corresponds to D = d′d:

(31) L(s, χ) = Ld′ (s)Ld(s). In particular, the Dedekind zeta function of K satisﬁes

(32) ζK(s) = L(s, 1) = ζ(s)LD(s). By (27) and well-known functional equations we get 2h(D) (33) L (0) = if D < 0 and L′ (0) = 1 h(D) log ϵ if D > 1. D w D 2 D The Hurwitz zeta function A useful tool to study Dirichlet L-functions at s = 0 is the Hurwitz zeta function, which is defined for x > 0 and Re(s) > 1 by ∑ ζ(s, x) = (n + x)−s. n≥0 For fixed x > 0 it has an analytic continuation in s to an entire function except for a simple pole at s = 1. It has the expansion due to Hurwitz [28] (see also p. 269 of [53]): ( ) Γ(1 − s) ∑ cos 2πnx ∑ sin 2πnx ζ(x, s) = sin πs + cos πs , (2π)1−s 2 n1−s 2 n1−s n≥1 n≥1 valid for 0 < x ≤ 1 and Re s < 0. This can be used to find the Laurent expansion of ζ(x, s) around s = 0. In particular we have 1 − ζ(x, 0) = 2 x, and from Kummer’s formula (10) ( ) − 1 ∂sζ(x, 0) = log (2π) 2 Γ(x) . Since for any D we have ∑|D| | |−s n LD(s) = D χD(n)ζ(s, |D| ) n=1 we immediately deduce the following for D ≠ 1: |D∑|−1 |D∑|−1 ( ) −| |−1 ′ − | | n (34) LD(0) = D nχD(n) and LD(0) = LD(0) log D + χD(n) log Γ |D| . n=1 n=1 − 1 ′ − 1 Also, for D = 1 we have ζ(0) = 2 , ζ (0) = 2 log 2π. If D > 1, we have that LD(0) = 0 and ∑D ( ) ′ − 1 nπ (35) LD(0) = 2 χD(n) log sin D , n=1 KRONECKER’S FIRST LIMIT FORMULA, REVISITED 9 − π by using the identity Γ(x)Γ(1 x) = sin x in the second formula of (34). Thus by (34), (35) and (33) we recover Dirichlet’s results |D∑|−1 ∑D ( ) − w − nπ h(D) = 2|D| nχD(n) and h(D) log ϵD = χD(n) log sin D n=1 n=1 for D < 0 and D > 1, respectively. The function − 2 (36) R(x) = ∂s ζ(x, 0) was studied by by Landau [32] and Ramanujan [37, Chapter 8] and applied to KLF by Deninger [10]. We have for D ≠ 1 |D∑|−1 ′′ − 2 | | − ′ | | − n (37) LD(0) = LD(0) log D 2LD(0) log D 2 χD(n)R( |D| ). n=1 KLF applied in the imaginary quadratic case Happily, KLF applies directly when D < 0. The examples we outlined above had K imaginary of class number 1 or 2. The general case when D < 0 and with a genus character is very similar. To start, 1 (38) ζ(s, A) = wζQ(s) for some positive integral Q of discriminant D. Here A is represented by a = Z + zQZ. Set zA = zQ for any such choice, for instance that zA ∈ F, the standard fundamental domain for Γ. This point zA is often called a CM point, the CM short for “complex multiplication”. It follows from (16) and the duplication formula for the gamma function that √ s −s 4 π ∗ | | 2 s s+1 (39) Λ(s, A) := D π Γ( 2 )Γ( 2 )ζ(s, A) = w E (zQ, s). Thus ζ(s, A) is entire but for a simple pole at s = 1 and by (20) satisfies the functional equation Λ(s, A) = Λ(1 − s, A). A direct application of KLF in the form (22) to (38) gives the next result. Theorem 2 (Kronecker). For D < 0 we have ζ(0,A) = − 1 and w ( ) ′ 2 − 1 − 1 | | 2 1 ζ (0,A) = w log 8π D + wH(zA). + ′ Given any character χ of ClD we can use this to give a formula for L (0, χ). In case χ is a genus character we obtain the following generalizations of the examples (11) and (14) from above.

Corollary 1. Suppose that D < 0. Then for w = wD the following holds: ∑ |D∑|−1 ( ) 1 2| | − w n (40) H(zA) = 2 h(D) log(16π D ) 2 χD(n) log Γ |D| (Lerch). ∈ + n=1 A ClD For D = d′d with d′ > 1 and d < 0 co-prime fundamental discriminants and χ the associated genus character we have ∑ wD ′ (41) χ(A)H(zA) = h(d)h(d ) log ϵd′ (Kronecker). wd ∈ + A ClD 10 W. DUKE, O.¨ IMAMOGLU,¯ AND A.T´ OTH´

Chowla and Selberg [8] reproved (40) independently and applied it to show that a period of an elliptic curve defined over an algebraic number field with CM by an order in K of discriminant D < 0 is an algebraic multiple of |D|−1 √ ∏ ( ) wχD(n) n 4h(D) π Γ |D| . n=1 Note that (41) can be put in the form of an average: ′ ∑ ′ 1 (h(d)/wd)(h(d ) log ϵd ) (42) h(D) χ(A)H(zA) = . h(D)/wD ∈ + A ClD Indefinite binary quadratic forms: Zagier’s limit formula From now on we assume that D > 0. Since real quadratic field norms comes from indefinite binary quadratic forms, KLF does not apply directly to compute ζ(s, A) when K is real quadratic. To address this problem, Zagier [54] gave an analogue of KLF for zeta functions associated to certain indefinite forms, which we will state here without proof. Let now Q(x, y) = ax2 + bxy + cy2 be a real binary quadratic form with positive coefficients and positive discriminant D = b2 − 4ac. Then the roots w′ < w of Q(1, −x) = 0 are positive. Define ∑ ∑ −s ZQ(s) = Q(n, m) . n≥1 m≥0

Theorem 3 (Zagier’s limit formula). The function ZQ(s) has an analytic continuation to 1 the half-plane Res > 2 with a simple pole at s = 1 and ( 1 w ) s 2 log w′ ′ lim D 2 ZQ(s) − = P (w, w ), s→1 s − 1 − y − π2 x − 1 − 1 x where P (x, y) = F(x) F(y) + Li2( x ) 6 + (log y )(γ0 2 log(x y) + 4 log y ) and ∑ ψ(nx) − log(nx) F(x) = , n n≥1 Γ′(x) where ψ(x) = Γ(x) is the digamma function. − 1 In fact, by [56, Satz 1, p.132] we have that ZQ(s) has an analytic continuation to Re s > 2 1 with a (possible) additional simple pole only at s = 2 and that 1 b b − (43) ZQ(0) = 24 ( a + c 6). ∈ + To proceed, we must relate ZQ(s) to ζ(s, A). Each A ClD contains fractional ideals of the form wZ + Z ∈ A where w ∈ K is such that w > wσ. Consider the minus (or backward) continued fraction of w: 1 (44) w = Ja , a , a ,...K = a − 0 1 2 0 1 a − 1 1 a2 − a3 − · · · where aj ∈ Z with aj ≥ 2 for j ≥ 1. This continued fraction is eventually periodic and has a unique primitive cycle ((n1, . . . , nℓ)) of length ℓ, only deﬁned up to cyclic permutations. KRONECKER’S FIRST LIMIT FORMULA, REVISITED 11

Diﬀerent admissible choices of w lead to the same primitive cycle. The continued fraction is purely periodic precisely when w is reduced in the sense that 0 < wσ < 1 < w

(see [54],[56]). The cycle ((n1, . . . , nℓ)) characterizes A; it is a complete class invariant. The length ℓ = ℓA, which is also the number of distinct reduced w, is another invariant as is the sum

(45) m = mA = n1 + ··· + nℓ. It is convenient to deﬁne a third class invariant

(46) Ψ(A) = mA − 3ℓA. Note that the cycle of A−1 is given by that of A reversed:

(47) ((nℓ, . . . , n1)) . To see this observe that A−1 is represented by (1/wσ)Z + Z and by [56, p.128] the continued fraction of 1/wσ has (47) as its cycle. Thus (48) Ψ(A−1) = Ψ(A). Similarly, using the relation between the minus continued fractions and regular simple continued fractions, it is shown in [27, p. 49] that

(49) Ψ(A) = ℓAJ − ℓA, √ where J denotes the class of the different ( D) of K. Hence we also have that Ψ(A−1J) = Ψ(AJ) = −Ψ(A), using the conjugacy invariance of Ψ, (48) and that J 2 = I, where I is the principal class. Let w1, w2, . . . , wℓ be the reduced values, which may be obtained from (44) and the cyclic permutations of the cycle. For each wk we define the indefinite binary quadratic form 1 σ Qk(x, y) = − σ (y + xwk)(y + xwk ), wk wk which has positive coefficients and discriminant one. Then Zagier gave the following important decomposition: ∑ℓ

(50) ζ(s, A) = ZQk (s). k=1

By evaluating the corresponding limit for each ZQk (s) we get a formula for the constant term s in the Laurent expansion of |D| 2 ζ(s, A) around s = 1 that involves a summation over the σ roots wk and wk of the ﬁxed function P from above. Theorem 4 (Zagier). For D > 0 ( ) ∑ℓ s log ϵD σ 2 − (51) lim D ζ(s, A) = P (wk, wk ). s→1 s − 1 k=1

Remarks: 12 W. DUKE, O.¨ IMAMOGLU,¯ AND A.T´ OTH´

1. To get the residue in (51), Zagier showed directly that

∏ℓ w (52) ϵ2 = k . D wσ k=1 k In particular, this together with (51) gives an apparently non-geometric proof of Dirichlet’s (27) when D > 1.

2. In a recent paper [49], Vlasenko and Zagier generalize (51) to s = 2, 3,... and interpret that in terms of the cohomology of Γ.

Hecke’s results Zagier’s paper followed earlier work of Hecke on the problem of extending Kronecker’s ideas to real quadratic fields. Hecke’s approach to KLF for a real quadratic field K was to integrate the definite version over appropriate cycles coming from the unit group of OK. His method is a direct descendent of that used by Dirichlet to prove his class number formula. A subtlety about real quadratic fields, which turns out to be crucial, is the possible existence of units with negative√ norm. As above we denote by I the principal class and by J the class of the different ( D) of K, which coincides with the class of principal ideals (α) where σ + O N(α) = αα < 0. Then ClD/J is the class group in the wide sense, which is trivial iff K is a UFD. Clearly J ≠ I iff OK contains no unit of norm −1. In this case each wide ideal class is the union of two narrow classes, say A and JA. A sufficient condition for J ≠ I is that D is divisible by a prime p ≡ 3 (mod 4). ∈ + Z Z ∈ σ S For a fixed narrow ideal class A ClD and a = w + A with w > w , let w be the σ geodesic in H with endpoints w and w. The modular closed geodesic CA on Γ\H is defined a b ∈ ∈ Z as follows. Define γw = ( c d ) Γ, where a, b, c, d are determined by

(53) ϵDw =aw + b

ϵD =cw + d, with ϵD our unit. Then γw is a primitive hyperbolic transformation in Γ with fixed points wσ and w. Since −2 −2 (cw + d) = ϵD < 1, we have that w is the attracting fixed point of γw. This induces on the geodesic Sw a clock- wise orientation. Distinct a and w for A induce Γ-conjugate transformations γw. If we choose some point z0 on Sw then the directed arc on Sw from z0 to γw(z0), when reduced modulo Γ, is the associated closed geodesic CA on Γ\H. It is well-defined for the class A and gives rise to a unique set of oriented arcs (that could overlap) in F. We also use CA to denote this set of arcs. It is not difficult to show that the closed geodesic CA−1J has the same image as CA but −1 with the opposite orientation. The arcs of CA retrace back over themselves when A J = A or, equivalently, A2 = J, i.e. J is in the principal genus. Sarnak [41] gave a nice account of these reciprocal geodesics. It is possible to compute γw associated to a reduced w as above simply using the ordering 1 1 of the cycle ((n1, . . . , nℓ)) that corresponds to w. In fact, if as usual we set T = ( 0 1 ) ,S = 0 1 ∈ ( −1 0 ) Γ then

n1 n2 nℓ (54) γw = T ST S ··· T S. KRONECKER’S FIRST LIMIT FORMULA, REVISITED 13

A nice treatment of this is given by S. Katok in [29, p.11.]. The class invariant Ψ(A) can now be computed in terms of the entries in γw. The Rademacher symbol is deﬁned for any a b ∈ γ = ( c d ) Γ by (55) Ψ(γ) = Φ(γ) − 3 sign(c(a + d)). ( ) Here Φ(γ) is the Dedekind symbol given for all γ = a b ∈ Γ by { c d b if c = 0 (56) Φ(γ) = d a+d − · ̸ c 12 sign c s(a, c) if c = 0, where s(a, c) is the Dedekind sum, deﬁned for gcd(a, c) = 1, c ≠ 0 by |c| (( ))(( )) ∑ n na s(a, c) = . c c n=1 As usual, ((x)) = 0 if x ∈ Z and otherwise ((x)) = x − ⌊x⌋ − 1/2. Rademacher showed that Ψ(γ) is invariant under conjugation in Γ. We also have that

(57) Ψ(A) = Ψ(γw). This follows from a well known formula for Ψ(γ) given by Rademacher [19] which, when applied to the expansion of γw in (54) gives Ψ(A) as defined in (46). + − Even and Odd characters. Say a character χ of ClD is even if χ(J) = 1 and odd if χ(J) = 1. In the more general terminology of [20], the odd characters are norm class characters of norm-signature type. Clearly the even characters are precisely those that induce wide class characters. It can be checked that a genus character coming from a decomposition D = d′d is odd if and only if d and d′ are both negative. In order to adapt (39) for the presence of an infinite groups of units, Hecke invented his famous trick of dividing out the action of the unit group on generators of principal ideals. For real quadratic fields this procedure amounts to either integrating the Eisenstein series with respect to arc length over the associated closed geodesic CA or integrating its derivative over this geodesic. In case I ≠ J it is necessary to define ( ) 1 ζ(s, A) = 2 ζ(s, A) ζ(s, JA) and their completions s − s − 2 s s 2 2 s s+1 2 (58) Λ+(s, A) = D π Γ( 2 ) ζ+(s, A) and Λ−(s, A) = D π Γ( 2 ) ζ−(s, A). In general ∑ L(s, χ) = χ(A)ζ(s, A) ∈ + A ClD where χ(J) = 1. Hecke found the following integral representations (see [44, Chap. II § 5 pp. 114–149.]): ∫ ∗ −1 (59) Λ+(s, A) = E (z, s)y |dz| C ∫ A ∗ (60) Λ−(s, A) = i ∂zE (z, s) dz. CA

These show, in particular, that s(1 − s)Λ(s, A) are entire and invariant under s 7→ 1 − s. First we examine the even case. By (59) and (21) we derive the following theorem. 14 W. DUKE, O.¨ IMAMOGLU,¯ AND A.T´ OTH´

Theorem 5 (Hecke). For D > 1 we have that ζ (0,A) = 0 and ∫ + ′ − 1 −1| | − 1 (61) ζ+(0,A) = 8 y dz = 4 log ϵD C ∫ A ( ) ′′ 1 1 D −1| | (62) ζ+(0,A) = 4 H(z) + 2 log( (2π)4 ) y dz . CA + ′ Given any even character χ of ClD we can use these to give a formulas for L (0, χ) and L′′(0, χ). In case χ is a genus character the resulting formulas imply the following. Recall the definition of R(x) from (36) and (37). Corollary 2. For D > 1 ∑ ∫ −1 (63) y |dz| = 2h(D) log ϵD and C ∈ + A A ClD ∑ ∫ D∑−1 −1| | n (64) H(z)y dz = (log D + 2 log 2π)h(D) log ϵD + 4 χD(n)R( D ). C ∈ + A n=1 A ClD For D = d′d with d′, d > 1 co-prime fundamental discriminants and χ the associated genus character, ∑ ∫ −1 ′ (65) χ(A) H(z)y |dz| = 2h(d)h(d ) log ϵd log ϵd′ . C ∈ + A A ClD Of course (63) is due to Dirichlet. The formulas (64) and (65) seem to have been first written down by Deninger [10] and Siegel [44, p.97.], respectively. ∫ In response to Hecke’s paper, Herglotz [25] managed to express the integral H(z)y−1|dz| CA in terms of integrals of elementary functions, but in general the resulting formula is pretty complicated. Before becoming aware of Herglotz’s papers, Zagier applied his more general formula (51) to ζ+(s, A) to give another (equivalent) version. As an interesting application of his formula, Herglotz applied it to (65) in special cases to evaluate some elementary integrals that seem to defy other proofs. Recently Muzaffar and Williams [35] found another equivalent formulation and gave several more examples, including ∫ √ 1 log(1 + x2+ 3) π2 √ √ dx = (1 − 3) + log 2 log(1 + 3). 0 1 + x 12 Finally, note that (65) can be put in the form ∫ ∑ ′ (h(d) log ϵ )(h(d ) log ϵ ′ ) ∑ ∫ 1 −1| | d d (66) −1 χ(A) H(z)y dz = , C y |dz| A A C h(D) log ϵD ∈ + A A ClD which is a real quadratic average analogous to (42).

Expansion of ζ−(s, A) around s = 0: the value ζ−(0,A)

From now on we consider only the remaining cases of ζ−(s, A) and L(s, χ) with odd genus characters χ. In view of the second formula of (58) and the functional equation we wish to ′ evaluate ζ−(0,A) and ζ−(0,A). As far as we know, of these only ζ−(0,A) has been evaluated before. We review this here ′ and consider ζ−(0,A) in the following section. There are several approaches to ζ−(0,A). Probably the most direct is that of Zagier in [56], where it is evaluated in terms of the KRONECKER’S FIRST LIMIT FORMULA, REVISITED 15 invariant Ψ of (46). Using (43), (49) and (50) it is elementary to deduce the following (see [56, Satz 2, p.132]): √ Theorem 6. For A a narrow ideal class in the real quadratic field K = Q( D) 1 ζ−(0,A) = 12 Ψ(A). + Given any odd character χ of ClD we can use this to give a formula for L(0, χ). In case χ is a genus character the resulting formula implies the following. Corollary 3. For D = d′d > 1 with d′, d < 0 co-prime fundamental discriminants and χ the associated genus character, ∑ 48h(d)h(d′) (67) χ(A)Ψ(A) = . wd wd′ ∈ + A ClD ≡ A nice special case√ is when D = 4q where q 3 (mod 4) is prime and h(4q) = 2, which is equivalent to Z[ q] being a UFD. This happens for the q of our first set of examples and, according to a well-known conjecture of Cohen and Lenstra [9], occurs for > 75% of all primes q ≡ 3 (mod 4). For these q > 3, (67) gives ∑ℓ − 1 − h( q) = 3 nk ℓ √ k=1 where ((n1, . . . , nℓ)) is the cycle of q. Alternatively, using (49), we have − 1 ′ − h( q) = 3 (ℓ ℓ), √ √ where ℓ′ is the length of the cycle of − q. For example, 163 has the cycle given by: ((5, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 4, 2, 2, 5, 26)) √ − 1 · − − so h( 163) = 3 108 35 = 1. One may check that the cycle of 163 has length 38. See [26] and [56] for more insight into these examples. Given (57) we can use Hecke’s representation (60) for another way to prove Theorem 6. The Fourier expansion of E∗(z, s) from (18) yields Hecke’s limit formula ∗ − πi ∗ lim ∂zE (z, s) = E2 (z), s→0 12 ∗ − 3 where E2 (z) = E2(z) and πy ∑ n E2(z) = 1 − 24 σ1(n)q . n≥1 Therefore by (58) and (60) we have ∫ 1 ∗ (68) ζ−(0,A) = 12 E2 (z) dz. CA (Now) apply Dedekind’s [10] evaluation of the transformation law for log η(z). For any γ = a b ∈ c d Γ we have − 1 − 2 πi (69) log η(γz) log η(z) = 4 log( (cz + d) ) + 12 Φ(γ), where Φ(γ) is given by the formula (56) and where we choose arg(−(cz + d)2) ∈ (−π, π). Since 1 1 d 12 E2(z) = πi dz log η(z), 16 W. DUKE, O.¨ IMAMOGLU,¯ AND A.T´ OTH´ a careful evaluation of the integral of − 1 shows that ∫ 4πy ∗ E2 (z) dz = Ψ(γw). CA This is in essence the first proof of Theorem 6 historically, modulo identity (57). Hecke [23, p. 416.] indicated how the evaluation of ζ−(0,A) depends on the transformation formula for log η(z) (see also [22], [36], [44, p.134.] and [55]). ′ We propose yet another proof of Theorem 6, one that will adapt well to computing ζ−(0,A) in terms of an integral of H(z). Let j(z) be the usual modular j-function as before. Write ′ 1 dj dxdy F j = 2πi dz and let dµ(z) = y2 and be the standard fundamental domain for Γ. The kernel function j′(τ) (70) K(z, τ) = , j(z) − j(τ) which is weight 0 in z and weight 2 in τ, shows up a lot in the theory of modular forms (see e.g. [1], [14]). It has the expansion, convergent for Im z < Im τ, ∑ m (71) K(z, τ) = jm(z)qτ m≥0 −m where qτ = e(τ) and jm(z) = q + O(q) is weakly holomorphic.

Figure 3. The function νI (z) for D = 12

Deﬁne ∫ j′(τ) (72) νA(z) = − dτ CA j(z) j(τ)

This function is Γ-invariant and is zero for Im z suﬃciently large. The value of νA(z) for z not on CA is an integer that counts with signs the number of crossings that a path from i∞ to z in F makes with CA. This is easily veriﬁed using the winding number and well-known properties of j. We claim that Theorem 6 is equivalent to the statement that ∫ 3 (73) π νA(z)dµ(z) = Ψ(A). F KRONECKER’S FIRST LIMIT FORMULA, REVISITED 17

This equivalence may be deduced from (68) and the following elegant formula, which is a consequence of Lemma 1 proven below and is obtained from it by taking s → 0 : ∫ ∗ 3 ′ 1 (74) E2 (τ) = π j (τ) dµ(z). F j(z) − j(τ) Here is an example illustrating the formula (73).√ Take D = 12 = (−3)(−4) as above with h(12) = 2. We have the reduced root wI = 2 + 3 = ϵ12 with wZ + Z = OK and with cycle ((4)) . Thus Ψ(I) = 4 − 3 = 1. In Figure 3 we have plotted CI , where the arcs are all oriented left to right and are given by the equations √ √ y = 3 − x2 and y = 3 − (x 1)2.

The values of νI (z) are also shown. It can be checked that (73) holds in this case since (75) ∫ ∫ ∫ √ 1/2 3−(x+u)2 3 − 6 π νI (z)dµ(z) = A(0) + A(1) + A( 1) = 1, where A(u) = π √ dµ(z). F 0 1−x2 In general, (73) was obtained in [5] for certain surfaces using topological arguments (Gauss- Bonnet) and adapted in [6] to Γ\H, but without using the analytic expression for νA(z) in (72). Partially motivated by their work and also by a desire to combine Zagier’s approach using continued fractions with the more geometric methods of Dirichlet/Hecke, in [15] we constructed a hyperbolic surface FA (an orbifold, actually) associated to A that is bounded by CA and has area ℓAπ, also a consequence of the Gauss-Bonnet theorem. This surface is a partial cover of F with mA − νA(z) points of FA over z ∈ F. In particular, we get a more analytic proof of (73) by combining this counting interpretation with (46). The Rademacher symbol has many other geometric/algebraic/topological interpretations. See Hirzebruch’s article [26] for relations to Hilbert modular surfaces. More recently, Ghys [18] gave the beautiful result that the Rademacher symbol gives the linking number of a modular knot (a lift of CA to the unit tangent bundle) with a certain trefoil knot. See also [16].

′ Evaluation of ζ−(0,A): new results ′ We now turn to the problem of computing ζ−(0,A), Returning to Hecke’s representation ′ (60), we see that to evaluate ζ−(0,A) using it directly we must integrate ∂zH2(z) from (26) over CA. Following the example of the third proof of Theorem 6 given above, we will instead integrate H(z) against νA(z) over F. The following result appears to be a new contribution to the classical theory of KLF. √ Theorem 7. For A a narrow ideal class in the real quadratic ﬁeld K = Q( D), where D = d′d with d and d′ negative co-prime fundamental discriminants, we have ∫ 1 ζ−(0,A) = 4π νA(z)dµ(z) and F∫ ( ) ′ − 1 1 D ζ−(0,A) = 4π νA(z) H(z) + 1 + 2 log( 28π4 ) dµ(z), F where νA(z) is deﬁned in (72). When applied to an odd genus character this yields an analogue of Lerch’s formula (40). 18 W. DUKE, O.¨ IMAMOGLU,¯ AND A.T´ OTH´

′ ′ ′ Corollary 4. Write h = h(d) and h = h(d ) and similarly w = wd and w = wd′ . For D = d′d with d′, d < 0 co-prime fundamental discriminants and χ the associated genus character, ∑ ∫ 3 48hh′ π χ(A) νA(z)dµ(z) = ww′ and F ∈ + A ClD ∑ ∫ ( |∑d|−1 |d∑′|−1 ) ′ ( ) ( ) 3 − h n h ′ n π χ(A) νA(z)H(z)dµ(z) = 24 w′ χd(n) log Γ |d| + w χd (n) log Γ |d′| F ∈ + n=1 n=1 A ClD 48hh′ 1 − 2 + ww′ ( 2 log D 1 + log(16π )). This corollary implies, after some computation, an identity that is somewhat analogous to (42) and (66) and is the main new result of this paper. It gives a surprising geometrical√ relationship√ between the CM points associated to the imaginary quadratic fields Q( d) and Q ′ ( d ) and the winding number√ functions√ νA(z) coming from the closed geodesics associated to the real quadratic field Q( D) = Q( d′d). Corollary 5. Assumptions as above, ∑ ∫ ∗ ∑ ∑ C∈Cl+ χ(C) F νC (z)H (z)dµ(z) ∑ D ∫ 1 ∗ 1 ∗ = h(d) H (zA) + h(d′) H (zB), C∈Cl+ χ(C) F νC (z)dµ(z) D A∈Cl+ B∈Cl+ d d′ where H∗(z) = H(z) − 1. In case d and d′ are prime discriminants there are exactly two genera, one containing I and the other containing J. If h(D) = 2, Corollary 5 simplifies to ∫ ∑ ∑ ∫ 1 ∗ 1 ∗ 1 ∗ (76) ν (z)dµ(z) νI (z)H (z)dµ(z) = h(d) H (zA) + h(d′) H (zB). F I F A∈Cl+ B∈Cl+ d d′ For example, when D = 12 we have numerically ∫ 3 ∗ − π H (z)νI (z)dµ(z) = B(0) + B(1) + B( 1) = 0.0882075 ..., F where ∫ ∫ √ 1/2 3−(x+u)2 6 ∗ B(u) = π √ H (z)dµ(z). 0 1−x2 By (76) and (75) this must equal ( ) √ 3Γ( 1 )6Γ( 1 )4 H∗(i) + H∗( 1+ −3 ) = −2 − log 3 4 , 2 512π7 as may be verified numerically. Given (73), the proof of Theorem 7 comes down to Hecke’s representation (60) and the following result. Lemma 1. For τ ∈ int F and K(z, τ) defined in (70) we have ∫ ∗ s(s−1) ∗ i∂τ E (τ, s) = 2 K(z, τ)E (z, s)dµ(z). F KRONECKER’S FIRST LIMIT FORMULA, REVISITED 19

Proof. Note that for ﬁxed τ ∈ F and y > Im τ + 1 we have that K(z, τ) = O(e−2πy) where as usual z = x + iy. Clearly 2 4y ∂z∂z = ∆ for ∆ from (24). Thus by (25) we need to show ∫

g(τ) = − K(z, τ)∂z g(z)dzdz, F ∗ after using that dzdz = 2idx dy and writing g(z) = ∂zE (z, s) for ﬁxed s ≠ 0, 1. For δ > 0, let Dτ (δ) = {z ∈ C : |z − τ| < δ} and F(Y ) = {z ∈ F : y ≤ Y }. Since K(z, τ) is holomorphic, for τ ∈ int F we need to show ∫ ( ) g(τ) = − lim ∂z g(z)K(z, τ) dzdz. Y →∞ F \ δ→0 (Y ) Dτ (δ) For τ ∈ int F choose Y large enough and δ small enough so we may apply Stokes’ theorem to get ∫ ∫ ∫ ( ) ∂z g(z)K(z, τ) dzdz = g(z)K(z, τ)dz − g(z)K(z, τ)dz. F(Y )\Dτ (δ) ∂FY ∂Dτ (δ) ∫ ∫ Now limY →∞ g(z)K(z, τ)dz = 0 and limδ→0 g(z)K(z, τ)dz = g(τ), giving the ∂FY ∂Dτ (δ) result. □

The main goal of our companion paper [15] was to study the distribution properties of the surfaces FA when A is averaged over a genus. For this we needed to make use of integrals F 1 over A of Maass cusp forms as well as the Eisenstein series E(z, s), but only for Res = 2 , since these are the eigenfunctions that occur in the spectral expansion of ∆. It is notable that H(z), which comes from s = 0, is not integrable on FA with respect to dµ. This is one reason for our use here of νA(z) and the singular kernel K(z, τ) in Lemma 1, instead of simply integrating H(z) over the surface FA.

4. Concluding remarks We have only looked at the applications of KLF to L-functions for quadratic K with genus characters. Already here there are interesting applications to certain quadratic extensions of K giving class number relations, but this is just the beginning, even when one is only interested in abelian extensions of quadratic fields (see [44]). A lot of attention has been devoted to finding elaborations of the ideas originated by Kronecker with his limit formula to other L-functions and other values of s. A rather different theme, one that we have emphasized without actually saying so, is the use of KLF to study averages of the height function H, like in (42), (66) and Corollary 5. The height function H(τ), thought of as being defined through KLF, gives the height of the torus C/L, where L = (Im τ)−1/2(Z + τZ). This definition has been generalized using spectral zeta functions of curves of genus greater than one. The corresponding higher dimensional versions of KLF and their applications to Riemann surfaces and in physics, pioneered in papers of Ray-Singer [38], Fay [17], D’Hoker–Phong [13] and Sarnak [39], [40], represent another major topic initiated by Kronecker’s ideas. 20 W. DUKE, O.¨ IMAMOGLU,¯ AND A.T´ OTH´ References

[1] Asai, Tetsuya; Kaneko, Masanobu; Ninomiya, Hirohito. Zeros of certain modular functions and an application. Comment. Math. Univ. St. Paul. 46 (1997), no. 1, 93–101. [2] A. Berger, Sur une application des nombres des classes des formes quadratiques binaires pour un deter- minant negatif. Nova Acta Soc. Sci. Upsaliensis (3) 12 (1883), pp. 29–31 [3] Birch, B. J.. Diophantine analysis and modular functions. 1969 Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) pp. 35–42 Oxford Univ. Press, London [4] Birch, B. J. Heegner’s proof. Arithmetic of L-functions, 281–291, IAS/Park City Math. Ser., 18, Amer. Math. Soc., Providence, RI, 2011. [5] Calegari, D.. SCL. MSJ Memoirs, 20. Mathematical Society of Japan, Tokyo, 2009. [6] Calegari, D. ; Louwsma, J.. Immersed surfaces in the modular orbifold, Proc. Amer. Math. Soc. 139 (2011), 2295–2308. [7] S. Chowla and A. Selberg. On Epstein’s zeta function (I), Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 371–374. [8] A. Selberg and S. Chowla. On Epstein’s zeta-function, J. Reine Angew. Math. 227 (1967), 86–110. [9] Cohen, H.; Lenstra, H. W., Jr. Heuristics on class groups. Number theory (New York, 1982), 26–36, Lecture Notes in Math., 1052, Springer, Berlin, 1984. [10] Deninger, Christopher, On the analogue of the formula of Chowla and Selberg for real quadratic fields. J. Reine Angew. Math. 351 (1984), 171–191. [11] Deuring, Max F. On Epstein’s zeta function. Ann. of Math. (2) 38 (1937), no. 3, 585–593. [12] Deuring, Max. Imaginärequadratische Zahlkörper mit der Klassenzahl Eins. Inventiones mathematicae 5.3 (1968): 169-179. [13] D’Hoker, E. and Phong D.H., On determinants of Laplacians on Riemann surfaces, Comm. Math Physics 104 537–545 (1986). [14] Duke, W.; Imamo¯glu, O.¨ Tóth, A.´ Cycle integrals of the j-function and mock modular forms. Ann. of Math. (2) 173 (2011), no. 2, 947–981. [15] Duke, W.; Imamo¯glu, O.;¨ Tóth, A.´ Geometric invariants for real quadratic fields. Ann. of Math. 184 (2016), no. 2, 94-990. [16] Duke, W.; Imamo¯glu, O.;¨ Tóth, A.´ Modular cocycles and linking numbers. Duke Math. J. 166 (2017), no. 6, 11791210. [17] Fay, John, Analytic torsion and Prym differentials. Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), pp. 107–122, Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, N.J., 1981. [18] Ghys,E., Knots and dynamics. International Congress of Mathematicians. Vol. I, 247–277, Eur. Math. Soc., Zürich, 2007. [19] Rademacher, H.; Grosswald, E., ”Dedekind Sums, The Carus Math.” Monographs, MAA (1972). [20] Hammond, W. F.; Hirzebruch, F., L-series, modular imbeddings, and signatures. Math. Ann. 204 (1973), 263–270. [21] E. Hecke, Uber¨ die Kroneckersche Grenzformell fürreelle quadratische Körper und die Klassenzhal relative-Abelscher Körper, Verhandlungen der Naturforschenden Gesellschaft in Basel Bd. 28, 1917, S.363– 372 = Werke 10, p. 198–207. [22] E. Hecke, Bestimmung der Klassenzahl einer neuen Reihe von algebraischen Zahlkörpern, Nachrichten der K. Gesellschaft der Wissenschafter zu Göttingen.Mathematische-physikalische Klasse 1921, S.1–23= Werke 15, p. 290–312 [23] E. Hecke, Darstellung von Klassenzahl als Peioden von Integralen3. Gattung aus dem Gebiet der elliptis- chen Modulfunktionen, Abhandlungen aus dem Mathematichen Seminar der Hamburgischen Universität Bd. 4, 1925, S.211–223=Werke 21, p. 405–417 [24] Heegner, Kurt. Diophantische analysis und modulfunktionen. Mathematische Zeitschrift 56.3 (1952): 227-253. [25] G. Herglotz, Uber¨ die Kroneckersche Grenzformel fürreele, quadratische Körper. I, II, Leipziger Ber. 75 (1923), p. 314 and p. 3137; Ges. Schriften, p. 466–484. [26] Hirzebruch, Friedrich E. P., Hilbert modular surfaces. Enseignement Math. (2) 19 (1973), 183–281. [27] Hirzebruch, F.; Zagier, D., Classification of Hilbert modular surfaces. Complex analysis and algebraic geometry, pp. 43–77. Iwanami Shoten, Tokyo, 1977. In Collected Papers II KRONECKER’S FIRST LIMIT FORMULA, REVISITED 21 ∑ [28] Hurwitz, A. ”Einige Eigenschaften der Dirichlet’schen Funktionen F (s) = (D/n)1/(ns), die bei der Bestimmung der Klassenanzahlen Binärerquadratischer Formen auftreten.” Z. fürMath. und Physik 27, 86-101, 1882. [29] Katok, Svetlana, Coding of closed geodesics after Gauss and Morse. Geom. Dedicata 63 (1996), no. 2, 123–145. [30] Kronecker, Werke IV, Chelsea (1929) p. 221–225. [31] Lagarias, Jeffrey C., and Rhoades, Robert C. Polyharmonic Maass forms for PSL(2, Z). The Ramanujan Journal 41.1-3 (2016): 191-232. [32] E. Landau, Uber¨ die zu einem algebraischen Zahlkörper gehörigeZetafunction und die Ausdehnung der Tschebyschefschen Primzahltheorie auf das Problem der Verteilung der Primideale J. reine angew. Math. (Crelles J.)125 (1902) 64-188. (Collected Works Vol. I, pp. 201–326) [33] Lang, Serge, Elliptic functions. With an appendix by J. Tate. Second edition. Graduate Texts in Math- ematics, 112. Springer-Verlag, New York, 1987. xii+326 pp. [34] Lerch M., Sur quelques formules relatives du nombre des classes, Bull. Sci. Math. 21 (1897) 290–304. [35] Muzaffar, Habib; Williams, Kenneth S., A restricted Epstein zeta function and the evaluation of some definite integrals. Acta Arith. 104 (2002), no. 1, 23–66. [36] Meyer, Curt, Die Berechnung der Klassenzahl Abelscher Körper über quadratischen Zahlkörpern. (Ger- man) Akademie-Verlag, Berlin, 1957. ix+132 pp. [37] Ramachandra, K., Some applications of Kronecker’s limit formulas. Ann. of Math. (2) 80 1964 104–148. [38] Ray, D. B.; Singer, I. M. Analytic torsion for complex manifolds. Ann. of Math. (2) 98 (1973), 154–177. [39] Sarnak, P., Special values of Selberg’s zeta-function. Number theory, trace formulas and discrete groups (Oslo, 1987), 457–465, Academic Press, Boston, MA, 1989. [40] Sarnak, Peter, Determinants of Laplacians; heights and finiteness. Analysis, et cetera, 601–622, Academic Press, Boston, MA, 1990. [41] Sarnak, Peter, Reciprocal geodesics. Analytic number theory, 217–237, Clay Math. Proc., 7, Amer. Math. Soc., Providence, RI, 2007. [42] Shintani, Takuro, On a Kronecker limit formula for real quadratic fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 167–199. [43] Shintani, Takuro, A proof of the classical Kronecker limit formula. Tokyo J. Math. 3 (1980), no. 2, 191–199. [44] Siegel, C. L., Advanced analytic number theory, Tata 1980 [45] Siegel, C. L., Bernoullische Polynome und quadratische Zahlkörper, Collected Papers IV, p. 9–40. (1968) [46] Stark, H. M., Class-number problems in quadratic fields. Actes du Congrés International des Mathématiciens(Nice, 1970), Tome 1, pp. 511–518. Gauthier-Villars, Paris, 1971. [47] Stark, H. M., Values of L-functions at s=1. I. L-functions for quadratic forms. Advances in Math. 7 1971 301–343 (1971). [48] Stark, Harold, The origin of the ”Stark conjectures”. Arithmetic of L-functions, 33–44, IAS/Park City Math. Ser., 18, Amer. Math. Soc., Providence, RI, 2011. [49] M. Vlasenko and D. Zagier, Higher Kronecker limit formulas for real quadratic fields J. reine angew. Math. 679 (2013), 23–64. [50] Weber, Heinrich Martin, Elliptische functionen und algebraische zahlen 1891 [51] Weber, Heinrich Martin, Lehrbuch der Algebra , vol 3 third ed. Chelsea 1961 [52] Weil, André,Elliptic functions according to Eisenstein and Kronecker. Reprint of the 1976 original. Classics in Mathematics. Springer-Verlag, Berlin, 1999. viii+93 pp. [53] Whittaker, E. T.; Watson, G. N., A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions. Fourth edition. Reprinted Cambridge University Press, New York 1962 vii+608 pp. [54] Zagier, Don, A Kronecker Limit formula for real quadratic fieldsds, Math. Ann, 213, (1975) 153–184 [55] Zagier, D., Nombres de classes et fractions continues JournésArithmétiquesde Bordeaux, Astérisque 24-25 (1975) 81-97 [56] Zagier, D. B., Zetafunktionen und quadratische Körper. (German) [Zeta functions and quadratic fields] Eine Einführungin die höhereZahlentheorie. [An introduction to higher number theory] Hochschultext. [University Text] Springer-Verlag, Berlin-New York, 1981. 22 W. DUKE, O.¨ IMAMOGLU,¯ AND A.T´ OTH´

UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555 E-mail address: [email protected] ETH, Mathematics Dept., CH-8092, Zurich,¨ Switzerland E-mail address: [email protected] Eotvos Lorand University, South Building Room 3.207, Budapest, Hungary E-mail address: [email protected]